Chapter 35: Diffraction and Polarization – Study Notes

Diffraction and Polarization

35.1 Diffraction by a Single Slit or Disk

Diffraction is a fundamental wave phenomenon where light bends around obstacles or passes through narrow openings, producing characteristic patterns of light and dark regions known as diffraction patterns. When light passes through a single slit, the resulting pattern consists of a broad central maximum flanked by narrower, dimmer maxima.

• Single Slit Diffraction: Each point in a narrow slit acts as a source of secondary wavelets (Huygens' principle), which interfere to form the observed pattern.

• Central Maximum: All wave fronts at angle $\theta = 0$ interfere constructively, producing a bright central band.

• Minima (Dark Bands): Occur when $D \sin\theta = m\lambda$, $m = \pm1, \pm2, \pm3, ...$

• Maxima (Bright Bands): Occur when $D \sin\theta = (m + 1/2)\lambda$, $m = \pm1, \pm2, \pm3, ...$

• Width of Central Maximum: The central maximum is twice as wide as the others: $2\lambda/D$.

Example: Light of wavelength 750 nm passes through a slit 1.0 x 10-3 mm wide. The width of the central maximum on a screen 20 cm away can be calculated using the above formulas.

35.2 Intensity in Single-Slit Diffraction Pattern

The intensity distribution in a single-slit diffraction pattern shows a strong central peak with rapidly decreasing intensity for subsequent maxima.

• Intensity vs. $\sin\theta$: The central maximum is the brightest and widest, with intensity dropping sharply for higher-order maxima.

Approximations in Diffraction Pattern

• For small angles, $\tan\theta \approx \sin\theta \approx \theta \approx y/D$

• Condition for Minimum: $a \sin\theta = m\lambda$

• Position of Minima on Screen: $y = \frac{m\lambda D}{a}$

Single and Double Slit Diffraction Patterns

Single-slit patterns have a broad envelope, while double-slit patterns show multiple sharp interference fringes within the envelope.

• Single Slit: Envelope pattern due to diffraction.

• Double Slit: Interference pattern modulated by the single-slit envelope.

35.5 Limits of Resolution; Circular Apertures

Diffraction through circular apertures is crucial in optical devices such as microscopes, telescopes, and cameras. The finite size of the aperture imposes a fundamental limit on the angular resolution of these instruments.

• Diffraction Pattern: Concentric circular bright and dark bands (Airy's disc).

• Minima: $\sin\theta = \pm1.22\lambda/D$, $\pm2.23\lambda/D$, $\pm3.24\lambda/D$, ...

• Maxima: $\sin\theta = \pm1.63\lambda/D$, $\pm2.68\lambda/D$, $\pm3.70\lambda/D$, ...

35.6 Resolution of Telescopes and Microscopes; the $\lambda$ Limit

The ability to distinguish two closely spaced objects is called resolution. Diffraction sets a limit to resolution, described by the Rayleigh criterion.

• Rayleigh Criterion: Two images are just resolvable when the center of one diffraction pattern is over the first minimum of the other.

• Resolution Limit (Telescopes): $\theta = \frac{1.22\lambda}{D}$ (in radians)

• Resolution Power (Microscopes): $RP = s = f\theta = \frac{1.22\lambda f}{D}$

• Lens Equation: $\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}$

• Magnification: $M = \frac{h_i}{h_o} = -\frac{d_i}{d_o}$

• Minimum Resolvable Detail: $RP \approx \frac{\lambda}{2}$

Example: Estimating the minimum separation between objects visible from an airplane, considering only diffraction and the diameter of the pupil.

35.8 Diffraction Grating

A diffraction grating consists of many equally spaced slits or lines, used to separate light into its component wavelengths with high precision.

• Types: Transmission grating (slits), Reflection grating (lines).

• Principal Maxima: $\sin\theta = \frac{m\lambda}{d}$, $m = 0, 1, 2, ...$

• Grating Spacing: $d = \frac{\text{length of grating}}{\text{number of lines}}$

• Effect of Number of Slits: More slits produce narrower and more intense maxima.

35.9 The Spectrometer and Spectroscopy

A spectrometer uses a diffraction grating or prism to measure wavelengths of light with high accuracy. Spectroscopy reveals the composition and properties of light sources.

• Measurement Principle: Wavelength determined by measuring the angle of diffraction: $\sin\theta = \frac{m\lambda}{d}$

• Types of Spectra:

  • Continuous Spectrum: Emitted by solids, liquids, and dense gases; all wavelengths present.

  • Emission Spectrum: Thin gases emit light at specific wavelengths (bright lines).

  • Absorption Spectrum: Thin gas absorbs light at specific wavelengths, producing dark lines.

Source	Spectrum Type	Description
Solid, liquid, dense gas	Continuous	All wavelengths, no gaps
Thin gas	Emission	Bright lines at specific wavelengths
Thin gas with background source	Absorption	Dark lines at specific wavelengths

35.11 X-Rays and X-Ray Diffraction

X-rays have very short wavelengths (1–100 nm), making conventional gratings unsuitable for diffraction experiments. Crystals, with their regular atomic spacing, act as natural diffraction gratings for X-rays.

• Bragg's Law: $2d \sin\theta = m\lambda$

• Application: Used to determine crystal structures, such as the double helix of DNA.

• Historical Note: Rosalind Franklin and Maurice Wilkins used X-ray diffraction to elucidate DNA structure.

Summary Table: Key Diffraction Equations

Phenomenon	Equation	Variables
Single-slit minima	$D \sin\theta = m\lambda$	$D$ = slit width, $\lambda$ = wavelength, $m$ = order
Circular aperture minima	$\sin\theta = 1.22\lambda/D$	$D$ = aperture diameter
Diffraction grating maxima	$\sin\theta = m\lambda/d$	$d$ = grating spacing
Bragg's Law (X-ray)	$2d \sin\theta = m\lambda$	$d$ = lattice spacing

Additional info:

• Diffraction and polarization are essential for understanding the limits of optical instruments and the analysis of light spectra.

• Rayleigh criterion is widely used in astronomy and microscopy to define the resolving power of instruments.

• X-ray diffraction is a cornerstone technique in crystallography and molecular biology.